Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
"Adjoint matrix" redirects here. For the transpose of cofactor, see
Adjugate matrix.
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an
complex matrix
is an
matrix obtained by transposing
and applying complex conjugation to each entry (the complex conjugate of
being
, for real numbers
and
). There are several notations, such as
or
,[1]
,[2] or (often in physics)
.
For real matrices, the conjugate transpose is just the transpose,
.
The conjugate transpose of an
matrix
is formally defined by
![{\displaystyle \left(\mathbf {A} ^{\mathrm {H} }\right)_{ij}={\overline {\mathbf {A} _{ji}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d446bde54031b1b3e6a0ee27d0891d7da946344) | | (Eq.1) |
where the subscript
denotes the
-th entry, for
and
, and the overbar denotes a scalar complex conjugate.
This definition can also be written as
![{\displaystyle \mathbf {A} ^{\mathrm {H} }=\left({\overline {\mathbf {A} }}\right)^{\operatorname {T} }={\overline {\mathbf {A} ^{\operatorname {T} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8a893002a447880e50c1611f380b9fa90ba437)
where
denotes the transpose and
denotes the matrix with complex conjugated entries.
Other names for the conjugate transpose of a matrix are Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix
can be denoted by any of these symbols:
, commonly used in linear algebra
, commonly used in linear algebra
(sometimes pronounced as A dagger), commonly used in quantum mechanics
, although this symbol is more commonly used for the Moore–Penrose pseudoinverse
In some contexts,
denotes the matrix with only complex conjugated entries and no transposition.
Suppose we want to calculate the conjugate transpose of the following matrix
.
![{\displaystyle \mathbf {A} ={\begin{bmatrix}1&-2-i&5\\1+i&i&4-2i\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6a52133d6cb6f1e12d676e8a2ed41065cad46b)
We first transpose the matrix:
![{\displaystyle \mathbf {A} ^{\operatorname {T} }={\begin{bmatrix}1&1+i\\-2-i&i\\5&4-2i\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd51f588910bf7ff1865023a73757ce321245202)
Then we conjugate every entry of the matrix:
![{\displaystyle \mathbf {A} ^{\mathrm {H} }={\begin{bmatrix}1&1-i\\-2+i&-i\\5&4+2i\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6cbe2a983b4661dfd8b8f0442e9526128f316a5)
A square matrix
with entries
is called
- Hermitian or self-adjoint if
; i.e.,
.
- Skew Hermitian or antihermitian if
; i.e.,
.
- Normal if
.
- Unitary if
, equivalently
, equivalently
.
Even if
is not square, the two matrices
and
are both Hermitian and in fact positive semi-definite matrices.
The conjugate transpose "adjoint" matrix
should not be confused with the adjugate,
, which is also sometimes called adjoint.
The conjugate transpose of a matrix
with real entries reduces to the transpose of
, as the conjugate of a real number is the number itself.
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by
real matrices, obeying matrix addition and multiplication:
![{\displaystyle a+ib\equiv {\begin{bmatrix}a&-b\\b&a\end{bmatrix}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c24779f45ee02c76388ba2ef1a2ccd4719c6763a)
That is, denoting each complex number
by the real
matrix of the linear transformation on the Argand diagram (viewed as the real vector space
), affected by complex
-multiplication on
.
Thus, an
matrix of complex numbers could be well represented by a
matrix of real numbers. The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an
matrix made up of complex numbers.
For an explanation of the notation used here, we begin by representing complex numbers
as the rotation matrix, that is,
Since
we are led to the matrix representations of the unit numbers as
A general complex number
is then represented as
The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.
[3]
Properties of the conjugate transpose
[edit]
for any two matrices
and
of the same dimensions.
for any complex number
and any
matrix
.
for any
matrix
and any
matrix
. Note that the order of the factors is reversed.[1]
for any
matrix
, i.e. Hermitian transposition is an involution.
- If
is a square matrix, then
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \operatorname{det}(A)}
denotes the determinant of
.
- If
is a square matrix, then
where
denotes the trace of
.
is invertible if and only if
is invertible, and in that case
.
- The eigenvalues of
are the complex conjugates of the eigenvalues of
.
for any
matrix
, any vector in
and any vector
. Here,
denotes the standard complex inner product on
, and similarly for
.
The last property given above shows that if one views
as a linear transformation from Hilbert space
to
then the matrix
corresponds to the adjoint operator of
. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis.
Another generalization is available: suppose
is a linear map from a complex vector space
to another,
, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of
to be the complex conjugate of the transpose of
. It maps the conjugate dual of
to the conjugate dual of
.